Logarithmic de Rham Cohomology

• Logarithmic de Rham cohomology is a framework that extends classical de Rham theory to spaces with logarithmic structures, such as fs log schemes and varieties with normal crossings divisors.
• It employs derived and prismatic methods to establish strong comparison theorems, clarify the degeneration of spectral sequences, and enhance p-adic Hodge theoretic insights.
• The theory integrates de Rham–Witt complexes, D-module techniques, and chiral de Rham models, offering practical implications for cycle theory, duality, and higher class field theory.

Logarithmic de Rham cohomology is a comprehensive theoretical framework extending the classical de Rham cohomology to spaces equipped with logarithmic structures, notably fine or fs log schemes and pairs consisting of a smooth variety with a normal crossings divisor. It appears as a central object in comparisons among topological, algebraic, and $p$-adic cohomology theories, facilitating a deeper understanding of cycle classes, period maps, and duality in positive and mixed characteristic. Recent advances introduce powerful derived and prismatic techniques, provide key inputs for $p$-adic Hodge theory, and clarify the Hodge-to-de Rham spectral sequence in the logarithmic context.

1. Logarithmic de Rham Complex: Construction and Basic Properties

Given a smooth algebraic variety $X$ over a field $k$ and a normal crossings divisor $D \subset X$, the logarithmic de Rham complex $\Omega_X^\bullet(\log D)$ is defined as the subcomplex of the meromorphic de Rham complex $\Omega_X^\bullet(*D)$ consisting of differential forms with at most simple poles along $D$ and whose differentials also have at most simple poles. Locally, if $(U; x_1,\ldots,x_n)$ are coordinates with $D = \{x_1 \cdots x_r = 0\}$, then

$$\Omega^1_U(\log D) = \bigoplus_{i=1}^r \mathcal{O}_U \cdot \frac{dx_i}{x_i} \oplus \bigoplus_{j=r+1}^n \mathcal{O}_U \cdot dx_j,$$

with higher degree forms defined by exterior powers. The logarithmic complex sits naturally as a subcomplex,

$$\Omega_X^\bullet(\log D) \subset \Omega_X^\bullet(*D).$$

The hypercohomology $\mathbb{H}^n(X,\Omega_X^\bullet(\log D))$ defines the logarithmic de Rham cohomology of $(X,D)$, denoted $H^n_{dR,\log}(X)$ (Bouali, 2023). This construction is functorial in morphisms of pairs and compatible with various filtrations, most notably the Hodge and conjugate filtrations.

2. Logarithmic de Rham–Witt and Derived Cohomologies

For fine log schemes over a perfect field of positive characteristic, the logarithmic de Rham–Witt complex $W_r\Omega_{X,\log}^\bullet$ provides a universal enhancement encoding both de Rham and crystalline information. This complex can be constructed via the Décalage operator $L\eta_p$ acting on pro-systems of log-crystalline complexes: $$W_r\Omega_{X,\log}^\bullet = H^\bullet\left(L\eta_p(R\Gamma_{r+1})\right), \qquad W\Omega_{X,\log}^\bullet = R\!\!\!\lim_r\, W_r\Omega_{X,\log}^\bullet.$$ It comes equipped with Frobenius ($F$), Verschiebung ($V$), and restriction ($R$) operators, satisfying the familiar Dwork–Cartier relations (Yao, 2018). For log-smooth schemes of log-Cartier type, this construction is naturally isomorphic to the Hyodo–Kato and Matsuue log de Rham–Witt complexes, preserving all additional structures.

The comparison with log-crystalline cohomology is provided by a canonical morphism

$$\tau: W\Omega_{X,\log}^\bullet \to A\Omega_X \otimes^L_{A_\text{inf}} W(k),$$

which is a quasi-isomorphism; at the $W_1$-level, it specializes to the log Cartier isomorphism.

The derived perspective, via log derived de Rham cohomology, uses the cotangent complex of log structures (Gabber–Illusie, Beilinson) and produces comparison theorems: $$dR_f^{\log} \xrightarrow{\simeq} R\Gamma_{\mathrm{crys}}((N \to B)/(M \to A))$$ for G-lci morphisms of prelog rings, with explicit control over filtrations (Hodge, conjugate) and induced spectral sequences (Bhatt, 2012).

3. Spectral Sequences, Hodge Theory, and Degeneration

The logarithmic de Rham complex admits a natural stupid (Hodge) filtration,

$$F^p\Omega_X^\bullet(\log D) = 0 \rightarrow \cdots \rightarrow 0 \rightarrow \Omega_X^p(\log D) \rightarrow \Omega_X^{p+1}(\log D) \rightarrow \cdots,$$

which induces a spectral sequence

$$E_1^{p,q} = H^q(X,\Omega_X^p(\log D)) \Rightarrow H^{p+q}_{dR,\log}(X).$$

A foundational result, due to Deligne–Illusie and extended by Kato and Hablicsek to the logarithmic case, asserts that this spectral sequence degenerates at $E_1$ provided $(X,D)$ lifts to $W_2(k)$ when $\operatorname{char}(k) = p > \dim X$. The degeneration is proved geometrically via twisted derived intersections and formality theorems for the Frobenius-pushforward of the log complex (Hablicsek, 2015).

4. Comparison Theorems, Prismatic Cohomology, and $p$-adic Hodge Theory

Logarithmic de Rham cohomology is a key player in $p$-adic comparison isomorphisms. For smooth rigid analytic spaces $X$ with strict simple normal crossings divisor $D$, there is a canonical comparison of log de Rham and $B_{dR}^+$-pro-étale cohomology,

$$H^i_{\mathrm{proet}}(X \setminus D, \mathbb{Z}_p) \otimes_{\mathbb{Z}_p} B_{dR}^+ \cong H^i_{dR}(X, D) \otimes_k B_{dR}^+,$$

compatible with filtrations and Galois actions. The logarithmic Hodge–de Rham spectral sequence degenerates and yields a Hodge–Tate-type decomposition (Li et al., 2018).

Logarithmic prismatic cohomology provides a further canonical bridge: for log-smooth schemes over bounded prelog prisms, there is a comparison

$R\Gamma_{\Prism}(X/A) \otimes_A^{L} A/I \cong R\Gamma_{dR}\bigl((X, M_X)/(A/I, M_A)\bigr).$

Natural filtrations (Nygaard and Hodge) coincide under the comparison; the graded pieces are log differentials, recovering Kato's Cartier isomorphism and the Hodge–Tate comparison (Koshikawa et al., 2023).

5. Logarithmic Classes, Cycle Theory, and Duality

The logarithmic subspace of de Rham cohomology is characterized as the image of $H^n(X, \Omega_X^\bullet(\log D))$ in $H^n_{dR}(X)$. Remarkably, for $X$ smooth over a field of characteristic zero, cycle classes of codimension $d$ are precisely those of Hodge type $(d,d)$ arising from logarithmic classes. Conversely, any such logarithmic class comes from an algebraic cycle, forming a bridge to the Tate and Hodge conjectures (Bouali, 2023). The p-adic analytic analogue, established via rigid-analytic spaces and formal models, mirrors the algebraic case.

In positive characteristic, logarithmic de Rham–Witt sheaves support the study of wild ramification and class field theory. A perfect duality is established: $$H^i(U, W_m\Omega_{X,\log}^r(D))_\mathrm{disc} \times \varprojlim_{D' \leq D} H^{d+1-i}(X, W_m\Omega_{X,\log}^{d-r}(D'))_\mathrm{pro} \to \mathbb{Z}/p^m\mathbb{Z}$$ for $X$ smooth proper over a finite field and $D$ an effective divisor, generalizing Serre–Grothendieck duality and underpinning higher dimensional class-field theory (Jannsen et al., 2016).

6. D-Module Theory and Logarithmic Comparison Theorems

The logarithmic comparison theorem (LCT) asks when the inclusion $\Omega_X^\bullet(\log D) \to \Omega_X^\bullet(*D)$ is a quasi-isomorphism. Using D-module theory, LCT holds for locally quasihomogeneous free divisors $D$ and is characterized by the isomorphism $$D_X \otimes_{V_X} \mathcal{O}_X(D) \simeq \mathcal{O}_X(*D)$$ in $D^b(D_X)$. For normal crossings divisors, explicit calculations confirm this, recovering topological cohomology. The failure of LCT for more general, non-quasihomogeneous free divisors highlights the subtlety of the Koszul property and the complexity of logarithmic stratifications (Castro-Jiménez et al., 2023).

7. Advanced Structures: Chiral de Rham, Log-Jet Spaces, and Extended Symmetries

The chiral de Rham complex admits a logarithmic counterpart $\Omega_X^{ch}(\log D)$, a sheaf of vertex algebras with weight-zero subspace $\Omega_X^\bullet(\log D)$. This enhancement encodes additional topological and superconformal structures, especially for log Calabi–Yau pairs. The $(q, y)$-character of its cohomology interpolates between the $\chi_y$-genus and Euler characteristic. Furthermore, the log chiral complex admits a birational description in terms of the sheaf of log forms on a suitably modified log-jet space, providing a geometric model for the interplay between jets, log structures, and vertex algebraic symmetry (Bouaziz, 6 Oct 2025).

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Selected References:

• "Logarithmic de Rham--Witt complexes via the Décalage operator" (Yao, 2018)
• "p-adic derived de Rham cohomology" (Bhatt, 2012)
• "Duality for relative logarithmic de Rham-Witt sheaves and wildly ramified class field theory over finite fields" (Jannsen et al., 2016)
• "Hodge theorem for the logarithmic de Rham complex via derived intersections" (Hablicsek, 2015)
• "Logarithmic Comparison Theorems" (Castro-Jiménez et al., 2023)
• "Logarithmic prismatic cohomology II" (Koshikawa et al., 2023)
• "Logarithmic jets and the chiral de Rham complex of a pair" (Bouaziz, 6 Oct 2025)
• "De Rham logarithmic classes and Tate conjecture" (Bouali, 2023)